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Knots and Primes pp 125–139Cite as

Homology Groups and Ideal Class Groups II—Higher Order Genus Theory

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Abstract

In this chapter we first study the 2-part of the 1st homology group of a double covering of the 3-sphere ramified over a link, introducing the higher order linking matrices which are defined by using the Milnor numbers of the link in Chap. 8. Imitating the method for a link, we study the 2-part of the narrow ideal class group of a quadratic extension of the rationals, using the arithmetic Milnor numbers introduced in Chap. 8. Our theorem may be regarded as a higher order generalization of Gauss’ and Rédei’s theorems on the 2-rank and 4-rank of the ideal class group.

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Authors and Affiliations

  1. Graduate School of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Masanori Morishita

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  1. Masanori Morishita
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Correspondence to Masanori Morishita .

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Morishita, M. (2012). Homology Groups and Ideal Class Groups II—Higher Order Genus Theory. In: Knots and Primes. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-2158-9_10

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